#8993693

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,5)(3,4,11,8)(6,13,14,9)(10,15,16,12)(17,18,22,26)(19,21,25,23)(20,24,28,32)(27,30,29,31)(33,34,39,43)(35,40,38,48)(36,37,42,41)(44,46,45,47);(1,3,7,11)(2,8,5,4)(6,10,14,16)(9,15,13,12)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,35,39,38)(34,40,43,48)(36,45,42,44)(37,47,41,46);(2,5)(4,8)(6,14)(10,16)(19,25)(20,28)(21,23)(24,32)(33,36,39,42)(34,41,43,37)(35,44,38,45)(40,47,48,46);(1,4)(2,3)(5,11)(6,12)(7,8)(9,10)(13,16)(14,15)(17,19,22,25)(18,23,26,21)(20,29,28,27)(24,30,32,31)(33,36)(34,37)(35,45)(38,44)(39,42)(40,47)(41,43)(46,48);(1,5,7,2)(3,8,11,4)(6,13,14,9)(10,15,16,12)(17,20,22,28)(18,24,26,32)(19,27,25,29)(21,30,23,31)(33,37)(34,42)(35,46)(36,43)(38,47)(39,41)(40,45)(44,48);(1,6)(2,9)(3,10)(4,12)(5,13)(7,14)(8,15)(11,16)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,29,32,27)(33,38,39,35)(34,40,43,48)(36,44,42,45)(37,47,41,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8900889);